We propose to extend Stein variational gradient descent (SVGD) [1]} to a multilevel SVGD (MLSVGD) that leverages the distributions from all levels \(\ell = 1, \dots , L\)  to more efficiently approximately sample from \(\pi \)  than traditional, single-level SVGD that uses the distribution \(\pi ^{(L)}\)  on the highest level \(L\)  only; thus, the proposed MLSVGD builds on the long history of exploiting hierarchies of discretizations in scientific computing (see below for literature review).
Our contributions are as follows: (1) an analysis that shows the cost complexity of the proposed MLSVGD is lower than the cost complexity of single-level SVGD; (2) a numerical algorithm that builds on an adaptive stopping criterion that can be applied in a black-box way; (3) numerical experiments with Bayesian inverse problems involving nonlinear diffusion-reaction and Euler-Bernoulli beam models that demonstrate that taking into account all levels \(\ell = 1, \dots , L\)  can lead to more than one order of magnitude speedup compared to single-level SVGD.
